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**D4.1 The language of probability**

Contents D4 Probability D2 D4.1 The language of probability D2 D4.2 The probability scale D2 D4.3 Calculating probability D2 D4.4 Probability diagrams D2 D4.5 Experimental probability

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**The language of probability**

Probability is a measurement of the chance or likelihood of an event happening. Describe the chance of drawing a red marble. even Chance متساوي الفرصة Unlikely غير مرجح Certain مؤكد Impossible مستحيل Likely مرجح Ask pupils to give examples of sentences for each phrase.

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The probability scale The chance of an event happening can be shown on a probability scale. Meeting with King Henry VIII A day of the week starting with a T The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles impossible Discuss the probability scale. The more likely an event is to occur, the further to the right of the line it is placed. The less likely an event is to occur, the further to the left. unlikely even chance likely certain Less likely More likely

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**Fair games A game is played with marbles in a bag.**

One of the following bags is chosen for the game. The teacher then pulls a marble at random from the chosen bag: bag b bag b bag a bag c Discuss the following questions: a) From which bag are the girls most likely to win a point? Why? b) From which bag are the boys least likely to win a point? Why? c) From which bag is impossible for the girls to win a point? d) From which bag are the boys certain to win a point? e) From which bag is it equally likely for the boys or the girls to win a point? f) Are any of the bags unfair? Why? If a red marble is pulled out of the bag, the girls get a point. If a blue marble is pulled out of the bag, the boys get a point. Which would be the fair bag to use?

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Fair games A game is fair if all the players have an equal chance of winning. Which of the following games are fair? A dice is thrown. If it lands on a prime number team A gets a point, if it doesn’t team B gets a point. There are three prime numbers (2, 3 and 5) and three non-prime numbers (1, 4 and 6). Remind pupils that 1 is not a prime number because it does have two different factors. Yes, this game is fair.

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Fair games Nine cards numbered 1 to 9 are used and a card is drawn at random. If a multiple of 3 is drawn team A gets a point. If a square number is drawn team B gets a point. If any other number is drawn team C gets a point. There are three multiples of 3 (3, 6 and 9). There are three square numbers (1, 4 and 9). Ask pupils to explain whether or not they think this game is fair. Although there are three cards that would give either team A or team B a point, there are four cards that would give team C a point. If appropriate, stress that the outcome of drawing a multiple of 3 and the outcome of drawing a square number are not mutually exclusive. It is possible to draw a card that is both a multiple of 3 and a square number, that is the card with a 9 on it. Therefore, P(A score a point) + P(B score a point) + P(C score a point) does not equal 1. Only the sum of all mutually exclusive outcomes equals 1. There are four numbers that are neither square nor multiples of 3 (2, 5, 7 and 8). No, this game is not fair. Team C is more likely to win.

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**Fair games A spinner has five equal sectors numbered 1 to 5.**

The spinner is spun many times. If the spinner stops on an even number team A gets 3 points. If the spinner stops on an odd number team B gets 2 points. 1 2 3 4 5 Suppose the spinner is spun 50 times. We would expect the spinner to stop on an even number 20 times and on an odd number 30 times. The game is fair because, although the spinner is less likely to stop on an even number, team A gets proportionally more points when it does. The probability of the spinner stopping on an even number is 0.4. The probability of the spinner stopping on an odd number is 0.6. So, the spinner is 50% more likely to stop on an odd number. The game is fair because team A gets 50% more points when the spinner stops on an even number. We can show that the game is fair by considering a theoretical game where the spinner is spun 50 times. Team A would score 20 × 3 points = 60 points Team B would score 30 × 2 points = 60 points Yes, this game is fair.

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**Bags of counters Choose a blue counter and win a prize!**

bag c bag a bag b bag c Discuss which bag is most likely to win. Stress that is the bag with the largest proportion of blue counters that is the most likely to win. Bag a has 4/12 blue counters, bag b has 3/10 blue counters and in bag c there are 2/5 blue counters. Converting these to decimals we have blue in bag a, 0.3 blue in bag b and 0.4 blue in bag c. Bag c is therefore the most likely to win the prize and then bag a. Bag b is the least likely to win. We can also look at the ratio of blue counters to yellow counters. In bag a there are two yellow counters for each blue counter. In bag b there are 21/3 yellow counters for each blue counter. In bag c there is 11/2 yellow counters for each blue counter. Bag c is the most likely to win because there are fewer yellow counters for each blue counter. Stress the difference between ratio and proportion. Proportion compares the parts to the whole and ratio compares the parts to each other. You are only allowed to choose one counter at random from one of the bags. Which of the bags is most likely to win a prize?

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The probability scale The chance of an event happening can be shown on a probability scale. Meeting with King Henry VIII A day of the week starting with a T The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles impossible Discuss the probability scale. The more likely an event is to occur, the further to the right of the line it is placed. The less likely an event is to occur, the further to the left. unlikely even chance likely certain Less likely More likely

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**The probability scale We measure probability on a scale from 0 to 1.**

If an event is impossible or has no probability of occurring then it has a probability of 0. If an event is certain it has a probability of 1. This can be shown on the probability scale as: 1 Advise pupils to always check that their answers giving probabilities are always between 1 and 0. Numbers greater than 1 cannot be used to describe probabilities. impossible even chance certain Probabilities are written as fractions, decimal and, less often, as percentages between 0 and 1.

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The probability scale Ask pupils to drag the pointer to the correct position on the scale.

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**D4.3 Calculating probability**

Contents D4 Probability D2 D4.1 The language of probability D2 D4.2 The probability scale D2 D4.3 Calculating probability D2 D4.4 Probability diagrams D2 D4.5 Experimental probability

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Higher or lower Start by revealing the first card and asking the class to predict whether the next card will be higher or lower? How often can pupils correctly predict whether the next card will be higher or lower? When can they be completely sure of their answer? Discuss strategies. Strategies may improve as you play more than once; for example, are pupils taking into account all the cards already turned over, or just the last one turned?

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**Listing possible outcomes**

When you roll a fair dice you are equally likely to get one of six possible outcomes: 1 6 1 6 1 6 1 6 1 6 1 6 Explain that the word ‘fair’ or ‘unbiased’ means that each outcome is equally likely. Some dice are ‘weighted’. That means that the weight of the dice is unevenly distributed and some numbers are more likely to appear than others. The probability of getting any number is 1 (certain) so the probability of getting each different number is 1 ÷ 6 or 1/6. Since each number on the dice is equally likely the probability of getting any one of the numbers is 1 divided by 6 or 1 6

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**Calculating probability**

What is the probability of the following events? 1) A coin landing tails up? 3) Drawing a seven of hearts from a pack of 52 cards? 1 2 P(tails) = 1 52 P(7 of ) = 2) This spinner stopping on the red section? 4) A baby being born on a Friday? For each example ask pupils to tell you the number of equally likely outcomes before revealing the probability. Introduce the notation of P(n) for the probability of an event n. 1 4 1 7 P(red) = P(Friday) =

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**Calculating probability**

If the outcomes of an event are equally likely then we can calculate the probability using the formula: Probability of an event = Number of successful outcomes Total number of possible outcomes For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. Point out that calculated probabilities are usually given as fractions but that they can also be given as decimals and (less often) as percentages. Ask pupils to give you the probabilities (as decimals, fractions and percentages) of getting a blue marble a red marble a yellow marble a purple marble a blue or a green marble, etc. What is the probability of pulling a green marble from the bag without looking? 3 10 P(green) = or 0.3 or 30%

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**Calculating probability**

This spinner has 8 equal divisions: What is the probability of the spinner landing on a red sector? a blue sector? a green sector? 2 8 = 1 4 a) P(red) = 1 8 b) P(blue) = 4 8 = 1 2 c) P(green) =

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**Calculating probability**

A fair dice is thrown. What is the probability of getting a 2? a multiple of 3? an odd number? a prime number? a number bigger than 6? an integer? 1 6 a) P(2) = 2 6 = 1 3 b) P(a multiple of 3) = 3 6 = 1 2 c) P(an odd number) =

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**Calculating probability**

A fair dice is thrown. What is the probability of getting a 2? a multiple of 3? an odd number? a prime number? a number bigger than 6? an integer? 3 6 = 1 2 d) P(a prime number) = Don’t write 6 e) P(a number bigger than 6) = 6 f) P(an integer) = = 1

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**Calculating probabilities**

Answer these questions giving each answer as a fraction or 0 or 1.

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**The probability of an event not occurring**

The following spinner is spun once: What is the probability of it landing on the yellow sector? 1 4 P(yellow) = What is the probability of it not landing on the yellow sector? Two probabilities that add up to one are sometimes called complementary probabilities (compare with number complements). 3 4 P(not yellow) = If the probability of an event occurring is p then the probability of it not occurring is 1 – p.

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**The probability of an event not occurring**

The probability of a factory component being faulty is What is the probability of a randomly chosen component not being faulty? P(not faulty) = 1 – 0.03 = 0.97 The probability of pulling a picture card out of a full deck of cards is What is the probability of not pulling out a picture card? 3 13 P(not a picture card) = 1 – = 3 13 10 13

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**The probability of an event not occurring**

The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring. Event Probability of the event occurring Probability of the event not occurring 3 5 2 5 A B 0.77 0.23 C 9 20 11 20 D 8% 92%

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**The probability of an event not occurring**

There are 60 sweets in a bag. 10 are cola bottles, 1 4 are fried eggs, 20 are hearts, the rest are teddies. What is the probability that a sweet chosen at random from the bag is: We can work out the number of sweets that are not teddies by finding the sum of 10, 20 and a ¼ of 60 to get 45. Modify the numbers to make this problem more challenging. 5 6 a) Not a cola bottle P(not a cola bottle) = 45 60 = 3 4 b) Not a teddy P(not a teddy) =

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**Adding mutually exclusive outcomes**

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability. For example, a game is played with the following cards: What is the probability that a card is a moon or a sun? 1 3 1 3 P(moon) = Ask pupils to tell you the probability of getting a crescent card or a star card. Reveal the solution on the board. Stress that only events that are mutually exclusive can be added in this way. For example, If we are drawing a card at random from a pack P(King) = 4/52, P(Club) = 13/52, but P(King or club) 4/ /52 because a card could be both a king and a club. and P(sun) = Drawing a moon and drawing a sun are mutually exclusive outcomes so, 1 3 + = 2 3 P(moon or sun) = P(moon) + P(sun) =

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**Adding mutually exclusive outcomes**

What is the probability that a card is yellow or a star? 1 3 1 3 P(yellow card) = and P(star) = Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star. Ask pupils to tell you the probability of getting a yellow card or a star card. Stress that this cannot be found by adding. The probability is 5/9 because one of the cards is both yellow and a star. P (yellow card or star) cannot be found simply by adding. We have to subtract the probability of getting a yellow star. 1 3 + – 1 9 = 3 + 3 – 1 9 = 5 9 P(yellow card or star) =

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**The sum of all mutually exclusive outcomes**

The sum of all mutually exclusive outcomes is 1. For example, a bag contains red counters, blue counters, yellow counters and green counters. P(blue) = 0.15 P(yellow) = 0.4 P(green) = 0.35 What is the probability of drawing a red counter from the bag? Explain that when we draw a counter from the bag it is either red, blue, yellow or green. These outcomes are therefore mutually exclusive, there are no other possible outcomes and so their combined probabilities must equal 1. Mutually exclusive outcomes can be added together. The decimals on this slide can be changed to make the problem more challenging. P(blue or yellow or green) = = 0.9 P(red) = 1 – 0.9 = 0.1

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**Finding all possible outcomes of two events**

Two coins are thrown. What is the probability of getting two heads? Before we can work out the probability of getting two heads we need to work out the total number of equally likely outcomes. There are three ways to do this: 1) We can list them systematically. Using H for heads and T for tails, the possible outcomes are: Stress that when there is more than one event it is important to list all the possible outcomes systematically. Listing the outcomes systematically means listing them in a logical order to make sure that none are left out. Explain that TH means, ‘a tail on the first coin and a head on the second’ and that HT means, ‘a head on the first coin and a tail on the second’. These are therefore two separate events. TH and HT are separate equally likely outcomes. TT, TH, HT, HH.

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**Finding all possible outcomes of two events**

2) We can use a two-way table. Second coin H T First coin HH HT TH TT From the table we see that there are four possible outcomes one of which is two heads so, 1 4 P(HH) =

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**Finding all possible outcomes of two events**

3) We can use a probability tree diagram. Outcomes Second coin HH H First coin H T HT T H TH T TT Again we see that there are four possible outcomes so, 1 4 P(HH) =

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**Finding the sample space**

A red dice and a blue dice are thrown and their scores are added together. What is the probability of getting a total of 8 from both dice? There are several ways to get a total of 8 by adding the scores from two dice. We could get a 2 and a 6, a 3 and a 5, a 4 and a 4, Stress that although we know that there are 5 different ways of getting a total of 8, we can’t find the probability of getting a total of 8 unless we know the total number of possible outcomes (the sample space). To find these we draw a two-way table as shown on the next slide. a 5 and a 3, or a 6 and a 2. To find the set of all possible outcomes, the sample space, we can use a two-way table.

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**Finding the sample space**

+ From the sample space we can see that there are 36 possible outcomes when two dice are thrown. 2 3 4 5 6 7 8 3 4 5 6 7 8 8 4 5 6 7 8 9 8 5 6 7 8 9 10 Five of these have a total of 8. As the two-way table is completed ask pupils what patterns they notice. Use the completed table to justify that the probability of the total score on the two dice being 8 is 5/36. Ask pupils to use the table to give the probabilities of other scores such as: P(3) P(a score less than 7) P(an even score) P(a score that is prime) P(a score that is square) Ask pupils to cancel down any fractions if possible. 8 6 7 8 9 10 11 5 36 P(8) = 8 7 8 9 10 11 12

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**D4.5 Experimental probability**

Contents D4 Probability D2 D4.1 The language of probability D2 D4.2 The probability scale D2 D4.3 Calculating probability D2 D4.4 Probability diagrams D2 D4.5 Experimental probability

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**Estimating probabilities based on data**

Suppose 1000 people were asked whether they were left- or right-handed. Of the 1000 people asked 87 said that they were left-handed. From this we can estimate the probability of someone being left-handed as or 87 1000 If we repeated the survey with a different sample the results would probably be slightly different. The more people we asked, however, the more accurate our estimate of the probability would be.

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Relative frequency The probability of an event based on data from an experiment or survey is called the relative frequency. Relative frequency is calculated using the formula: Relative frequency = Number of successful trials Total number of trials For example, Ben wants to estimate the probability that a piece of toast will land butter-side-down. He drops a piece of toast 100 times and observes that it lands butter-side-down 65 times. Relative frequency = 65 100 = 13 20

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Relative frequency Sita wants to know if her dice is fair. She throws it 200 times and records her results in a table: Number Frequency Relative frequency 1 31 2 27 3 38 4 30 5 42 6 32 31 200 = 0.155 27 200 = 0.135 38 200 = 0.190 30 200 = 0.150 If the dice were fair we would expect to get each outcome an equal number of times. Stress that in an experiment the results are random and unpredictable. If we repeated this experiment we would get a different set of results. Conclude that this dice seems to be fair because the relative frequencies are all close to 1/6 or 42 200 = 0.210 32 200 = 0.160 Is the dice fair?

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**Experimental probability**

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Expected frequency The theoretical probability of an event is its calculated probability based on equally likely outcomes. If the theoretical probability of an event can be calculated, then when we do an experiment we can work out the expected frequency. Expected frequency = theoretical probability × number of trials If you rolled a dice 300 times, how many times would you expect to get a 5? The theoretical probability of getting a 5 is . 1 6 So, expected frequency = × 300 = 1 6 50

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Expected frequency If you tossed a coin 250 times how many times would you expect to get a tail? 1 2 Expected frequency = × 250 = 125 If you rolled a fair dice 150 times how many times would you expect to a number greater than 2? Stress that the greater the number of trials the closer the experimental frequency will be to the expected frequency. 2 3 Expected frequency = × 150 = 100

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Spinners experiment Use the spinners experiment to compare the theoretical probability with the relative frequency for each spinner. Notice that the more times the spinner is spun the closer the relative frequency gets to the theoretical probability.

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**Worksheet ( 1 ) Zayed althani school Math department Coin 2 H T Coin 1**

Mohamad badawi : * Write the sample space ( all possible results ) when rolling a fair dice اكتب فضاء العينة ( مجموعة جميع النواتج الممكنة ) عند إلقاء حجر نرد منتظم Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : 1) The upper face is a number greater than 5. ………………. ظهور عدد اكبر من 5 على الوجه العلوي 2) The upper face is a prime number. ………………. ظهور عدد أولي على الوجه العلوي Coin 2 H T Coin 1 * You toss 2 coins together ألقيت قطعتي نقود معاً 1) Complete the table to show all possible results . أكمل الجدول لتبين جميع النواتج الممكنة Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : 1) You will get 2 heads ………………. ستحصل على صورتين 2) At least one head ………………. ستظهر صورة واحدة على الأقل 3) You will get one tail exactly………………. ستظهر الكتابة مرة واحدة بالضبط

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**2 cards were randomly drawn from a deck of 52 cards**

Worksheet ( 2 ) Zayed althani school Math department Mohamad badawi : 2 cards were randomly drawn from a deck of 52 cards تم سحب ورقتين من علبة لعب الورق ( الشدة ) التي تحوي 52 ورقة 1) Complete the table to show all possible results . أكمل الجدول لتبين جميع النواتج الممكنة Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : Card 2 Card 1 1) The 2 cards are of the same color. ………………. البطاقتان من نفس اللون 2) The 2 cards are spades. ………………. البطاقتان من نوع البستوني 3) One of the cards was green. ………………. احد البطاقتين خضراء : Spade بستوني : Clubs سباتي : Diamond ديناري : Heart كبه ( قلب ) 4) The 2 cards are either red or black or red and black………………. البطافتان إما حمراوتان أو سوداوتان أو حمراء وسوداء 5) At least one of the 2 cards wasn’t a picture. ………………. على الأقل احدهما ليست صورة

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Bell Work/Cronnelly. A= 143 ft 2 ; P= 48 ft A= 2.3 m; P= 8.3 m A= 292.5 ft 2 ; P= 76 ft 2/12; 1/6 1/12 8/12; 2/3 6/12; 1/2 0/12 4/12; 1/3 5/12 6/12; 1/2.

Bell Work/Cronnelly. A= 143 ft 2 ; P= 48 ft A= 2.3 m; P= 8.3 m A= 292.5 ft 2 ; P= 76 ft 2/12; 1/6 1/12 8/12; 2/3 6/12; 1/2 0/12 4/12; 1/3 5/12 6/12; 1/2.

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